SOLUTION: Hi I need help solving the system of equations by graphing y=-3x+3 5x+y=5 Please use method of substitution and elimination to solve (use both please). Please show me your st

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Question 145822: Hi I need help solving the system of equations by graphing
y=-3x+3
5x+y=5
Please use method of substitution and elimination to solve (use both please). Please show me your steps because I keep doing this wrong.
Thanks so much for your help

Answer by jim_thompson5910(35256)   (Show Source): You can put this solution on YOUR website!
Substitution:


Start with the given system





Plug in into the first equation. In other words, replace each with . Notice we've eliminated the variables. So we now have a simple equation with one unknown.


Combine like terms on the left side


Subtract 3 from both sides


Combine like terms on the right side


Divide both sides by 2 to isolate x



Divide




Now that we know that , we can plug this into to find



Substitute for each


Simplify


So our answer is and which also looks like



Notice if we graph the two equations, we can see that their intersection is at . So this verifies our answer.


Graph of (red) and (green)







Elimination:

Start with the first equation


Add 3x to both sides




So we now have the system of equations:





Now in order to solve this system by using elimination/addition, we need to solve (or isolate) one variable. I'm going to solve for y.





In order to solve for one variable, we must eliminate the other variable. So if we wanted to solve for , we would have to eliminate (or vice versa).


So lets eliminate . In order to do that, we need to have both coefficients that are equal in magnitude but have opposite signs (for instance 2 and -2 are equal in magnitude but have opposite signs). This way they will add to zero. By adding to zero, they can be eliminated.



So to make the coefficients equal in magnitude but opposite in sign, we need to multiply both coefficients by some number to get them to an common number. So if we wanted to get and to some equal number, we could try to get them to the LCM.



Since the LCM of and is , we need to multiply both sides of the top equation by and multiply both sides of the bottom equation by like this:




Multiply the top equation (both sides) by
Multiply the bottom equation (both sides) by




Distribute and multiply





Now add the equations together. In order to add 2 equations, group like terms and combine them



Combine like terms and simplify



Notice how the x terms cancel out




Simplify




Divide both sides by to isolate y




Reduce



Now plug this answer into the top equation to solve for x

Start with the first equation



Plug in




Multiply



Subtract 0 from both sides


Combine like terms on the right side


Divide both sides by 3 to isolate x



Divide




So our answer is
and



which also looks like




Now let's graph the two equations (if you need help with graphing, check out this solver)


From the graph, we can see that the two equations intersect at . This visually verifies our answer.




graph of (red) and (green) and the intersection of the lines (blue circle).

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